Growth of Perturbations

We will follow the explanation in the CCL paper (Chisari et al. 2019) to compute the growth rate and the growth factor. In short, we want to solve the following differential equation:

\[ \dfrac{d}{da}\left(a^{3}H(a)\dfrac{dD}{da}\right) = \dfrac{3}{2}\Omega_{m}(a)aH(a)D(a) \]

where

  • \(a\) is the scale factor,
  • \(H(a)\) is the Hubble parameter at scale factor \(a\),
  • \(D(a)\) is the linear growth factor of matter perturbation and
  • \(\Omega_{m}(a)\) is the matter density at scale factor \(a\).

Following the CCL paper, we define \(D(a) = ag(a)\) and hence, \(D'(a) = ag'(a) + g(a)\). The notation \('\) denotes the derivative of the quantity with respect to \(a\). Working through the maths, we can show:

\[ g''(a) + \left(\dfrac{5}{a}+\dfrac{H'(a)}{H(a)}\right)g'(a) - \dfrac{1}{a^{2}}\left(\dfrac{3}{2}\Omega_{m}(a)-a\dfrac{H'(a)}{H(a)}-3\right)g(a) = 0 \]

We can also write the above expression in terms of \(E(a)\), that is,

\[ \dfrac{H'(a)}{H(a)} = \dfrac{E'(a)}{E(a)} \]

Next, we define \(\alpha(a)\) and \(\beta(a)\) as follows:

\[ \begin{split} \alpha(a) &=\dfrac{5}{a} + \dfrac{H'(a)}{H(a)} \\ \beta(a) &=-\dfrac{1}{a^{2}}\left(\dfrac{3}{2}\Omega_{m}(a) - a\dfrac{H'(a)}{H(a)}-3\right) \end{split} \]

and we now have

\[ g''(a) + \alpha(a)g'(a) + \beta(a)g(a) = 0. \]

At very high redshift, \(g(a=0)=1\) and \(g'(a=0)=0\). If we define:

\[ \boldsymbol{y} =\left(\begin{array}{c} y_{0}(a)\\ y_{1}(a)\\ \end{array}\right)= \left(\begin{array}{c} g(a)\\ g'(a)\\ \end{array}\right) \]

and

\[ \boldsymbol{y}' =\left(\begin{array}{c} y'_{0}(a)\\ y'_{1}(a)\\ \end{array}\right)=\left(\begin{array}{cc} 0 & 1\\ -\beta(a) & -\alpha(a) \end{array}\right)\,\left(\begin{array}{c} y_{0}(a)\\ y_{1}(a) \end{array}\right) \]

The above differential equation can then be solved used various numerical methods such as the Euler's method, Runge-Kutta \(4^{th}\) order method, Cash-Karp method and others. Once we have \(g(a)\), we can compute:

  • the normalised linear growth factor:
\[ D(a) = \dfrac{ag(a)}{g(a=1)} \]
  • the logarithmic growth rate:
\[ f(a)\equiv \dfrac{d\,\textrm{ln}D}{d\,\textrm{ln}a}=\dfrac{aD'}{D} = 1+a\dfrac{g'(a)}{g(a)} \]